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Sei $a\in \intgr$ beliebig. \textbf{Zu zeigen:} $a\sim a$.\\ Offensichtlich gilt $\modfn(a,n)=\modfn(a,n)$.\\ - Per Konstruktion gilt also $(a,b)\sim(a,b)$. + Per Konstruktion gilt also $a\sim a$. \item[\uwave{{\itshape Symmetrie:}}] Seien $a, a'\in \intgr$ beliebig. @@ -3427,13 +3428,13 @@ für $a,b\in\intgr$. \,\text{per Definition}\\ &= &\{a\in\intgr \mid \modfn(a,n)=\modfn(k,n)\}\\ &= &\{a\in\intgr \mid \modfn(a,n)=k\}\\ - &= &\{a\in\intgr \mid \exists{q\in\intgr:~}a=qn+r\}\\ + &= &\{a\in\intgr \mid \exists{q\in\intgr:~}a=qn+k\}\\ &= &\{qn+r \mid q\in\intgr\}\\ - &= &\intgr\cdot n + r.\\ + &= &\intgr\cdot n + k.\\ \end{mathe} Also lassen sich die Äquivalenzklassen durch die Teilmengen - ${\{\intgr\cdot n+r\mid r\in\{0,1,\ldots,n-1\}\}}$ + ${\{\intgr\cdot n+k\mid k\in\{0,1,\ldots,n-1\}\}}$ darstellen. \end{enumerate}